Nomenclature

CAVITATION AND BUBBLE DYNAMICS

by Christopher Earls Brennen © Oxford University Press 1995

ROMAN LETTERS

a	Amplitude of wave-like disturbance
A	Cross-sectional area or cloud radius
b	Body half-width
B	Tunnel half-width
c	Concentration of dissolved gas in liquid, speed of sound, chord
c_k	Phase velocity for wavenumber k
c_P	Specific heat at constant pressure
C_D	Drag coefficient
C_L	Lift coefficient
,	Unsteady lift coefficients
C_M	Moment coefficient
,	Unsteady moment coefficients
C_ij	Lift/drag coefficient matrix
C_p	Coefficient of pressure
C_pmin	Minimum coefficient of pressure
d	Cavity half-width, blade thickness to spacing ratio
D	Mass diffusivity
f	Frequency in Hz.
f	Complex velocity potential, φ+iψ
f_N	A thermodynamic property of the phase or component, N
Fr	Froude number
g	Acceleration due to gravity
g_x	Component of the gravitational acceleration in direction, x
g_N	A thermodynamic property of the phase or component, N
(f)	Spectral density function of sound
h	Specific enthalpy, wetted surface elevation, blade tip spacing
H	Henry's law constant
Hm	Haberman-Morton number, normally gμ⁴/ρS³
i,j,k	Indices
i	Square root of -1 in free streamline analysis
I	Acoustic impulse
I^*	Dimensionless acoustic impulse, 4πI {\cal R} / ρ_L U_∞ R_H²
I_Ki	Kelvin impulse vector
j	Square root of -1
k	Boltzmann's constant, polytropic constant or wavenumber
k_N	Thermal conductivity or thermodynamic property of N
K_G	Gas constant
K_ij	Added mass coefficient matrix, 3M_ij/4ρπR³
Kc	Keulegan-Carpenter number
Kn	Knudsen number, λ/2R
ℓ	Typical dimension in the flow, cavity half-length
L	Latent heat of vaporization
m	Mass
m_G	Mass of gas in bubble
m_p	Mass of particle
M_ij	Added mass matrix
n	Index used for harmonics or number of sites per unit area
N(R)	Number density distribution function of R
	Cavitation event rate
Nu	Nusselt number
p	Pressure
p_a	Radiated acoustic pressure
p_s	Root mean square sound pressure
p_S	A sound pressure level
p_G	Partial pressure of gas
P	Pseudo-pressure
Pe	Peclet number, usually WR/α_L
q	Magnitude of velocity vector
q_c	Free surface velocity
Q	Source strength
r	Radial coordinate
R	Bubble radius
R_B	Equivalent volumetric radius, [3τ/4π]^1/3
R_H	Headform radius
R_M	Maximum bubble radius
R_N	Cavitation nucleus radius
R_P	Nucleation site radius
	Distance to measurement point
Re	Reynolds number, usually 2WR/ν_L
s	Coordinate measured along a streamline or surface
s	Specific entropy
S	Surface tension
St	Strouhal number, 2fR/W
t	Time
t_R	Relaxation time for relative motion
t_*	Dimensionless time, t/t_R
T	Temperature
u,v,w	Velocity components in cartesian coordinates
u_i	Velocity vector
u_r,u_θ	Velocity components in polar coordinates
u′	Perturbation velocity in x direction, u-U_∞
U, U_i	Fluid velocity and velocity vector in absence of particle
V, V_i	Absolute velocity and velocity vector of particle
U_∞	Velocity of upstream uniform flow
w	Complex conjugate velocity, u-iv
w	Dimensionless relative velocity, W/W_∞
W	Relative velocity of particle
W_∞	Terminal velocity of particle
We	Weber number, 2ρW²R/S
z	Complex position vector, x+iy


GREEK LETTERS

α	Thermal diffusivity, volume fraction, angle of incidence
β	Cascade stagger angle, other local variables
γ	Ratio of specific heats of gas
Γ	Circulation, other local parameters
δ	Boundary layer thickness or increment of frequency
δ_D	Dissipation coefficient
δ_T	Thermal boundary layer thickness
ε	Fractional volume
ζ	Complex variable, ξ+iη
η	Bubble population per unit liquid volume
η	Coordinate in ζ-plane
θ	Angular coordinate or direction of velocity vector
κ	Bulk modulus of compressibility
λ	Mean free path of molecules or particles
Λ	Accommodation coefficient
μ	Dynamic viscosity
ν	Kinematic viscosity
ξ	Coordinate in ζ-plane
	Logarithmic hodograph variable, χ+iθ
ρ	Density
σ	Cavitation number
σ_c	Choked cavitation number
σ_ij	Stress tensor
Σ	Thermal parameter in bubble growth
τ	Volume of particle or bubble
ø	Velocity potential
ø′	Acceleration potential
φ	Fractional perturbation in bubble radius
Φ	Potential energy
χ	log(q_c/\|w\|)
ψ	Stream function
ω	Radian frequency
ω^*	Reduced frequency, ωc/U_∞


SUBSCRIPTS

On any variable, Q:

Q_o	Initial value, upstream value or reservoir value
Q₁,Q₂,Q₃	Components of Q in three Cartesian directions
Q₁,Q₂	Values upstream and downstream of a shock
Q_∞	Value far from the bubble or in the upstream flow
Q_B	Value in the bubble
Q_C	Critical values and values at the critical point
Q_E	Equilibrium value or value on the saturated liquid/vapor line
Q_G	Value for the gas
Q_i	Components of vector Q
Q_ij	Components of tensor Q
Q_L	Saturated liquid value
Q_n	Harmonic of order n
Q_P	Peak value
Q_S	Value on the interface or at constant entropy
Q_V	Saturated vapor value
Q_*	Value at the throat


SUPERSCRIPTS AND OTHER QUALIFIERS

On any variable, Q:

	Mean value of Q or complex conjugate of Q
	Complex amplitude of oscillating Q
	Laplace transform of Q(t)
	Coordinate with origin at image point
	Rate of change of Q with time
	Second derivative of Q with time
Q⁺,Q^-	Values of Q on either side of a cut in a complex plane
δQ	Small change in Q
Re{Q}	Real part of Q
Im{Q}	Imaginary part of Q

UNITS

In most of this book, the emphasis is placed on the nondimensional parameters that govern the phenomenon being discussed. However, there are also circumstances in which we shall utilize dimensional thermodynamic and transport properties. In such cases the International System of Units will be employed using the basic units of mass (kg), length (m), time (s), and absolute temperature (K); where it is particularly convenient units such as a joule (kg m²/s²) will occasionally be used.